. Inference with Rewriting Rules (1) . .. NAKAMURA Masaki . Toyama Pref. Univ. December 17, 2013 NAKAMURA Masaki (Toyama Pref. Univ.) Inference with Rewriting Rules (1) December 17, 2013 1 / 22 . Content 1 Term rewriting systems .. 2 Termination Semantic methods Syntactic methods Incremental proofs of termination .. NAKAMURA Masaki (Toyama Pref. Univ.) Inference with Rewriting Rules (1) December 17, 2013 2 / 22 . Equational reasoning by rewriting The CafeOBJ reduction command implements equational logic by rewriting and can be used as a powerful interactive theorem proving system { { X + 0 = X (1) X + 0 ! X (1) E = R = X + s(Y ) = s(X + Y ) (2) X + s(Y ) ! s(X + Y ) (2) Bidirectional equations are not suitable Equations are regarded as left-to-right for automated equational reasoning rewrite rules s(0) + s(s(0)) =2+ s(s(0) + s(0)) s(0) + s(s(0)) =?E s(s(0)) + s(0) # # =2+ s(s(s(0) + 0)) 2 2 =1+ s(s(s(0))) s(s(0) + s(0)) s(s(s(0)) + 0) # # =1− s(s(s(0)) + 0) 2 1 ! =2− s(s(0)) + s(0) s(s(s(0) + 0)) 1 s(s(s(0))) NAKAMURA Masaki (Toyama Pref. Univ.) Inference with Rewriting Rules (1) December 17, 2013 3 / 22 . How to describe equational specifications Not all equational specifications can work well as rewriting systems R0 = fa ! b; b ! ag may result in an infinite loop ! ! ! ··· a R0 b R0 a R0 R1 = fa ! b; a ! cg cannot prove b = c by rewriting ! 6! 6! b R1 a R1 c; but c R1 a; b R1 a Unfortunately, CafeOBJ system does not find suitable rewrite rules (directions) from input equational specifications. We need a way to describe equational specifications as rewrite systems. NAKAMURA Masaki (Toyama Pref. Univ.) Inference with Rewriting Rules (1) December 17, 2013 4 / 22 . Terms To concentrate to understand a mechanism of rewriting, we first assume simple equational specifications which has only one sort, operators without any attributes (assoc, comm, etc) and unconditional equations Σ : the set of operators f ; g; h;::: 2 Σ TΣ(X ) (abbr. T ): the set of terms constructed from operators in Σ and variables in the variable set X X ; Y ; Z;::: 2 X f (0); g(c; X ); h(f (0); g(c; X ); Y );::: 2 TΣ(X ) NAKAMURA Masaki (Toyama Pref. Univ.) Inference with Rewriting Rules (1) December 17, 2013 5 / 22 . Term rewriting systems A (Σ-)rewrite rule is defined as a pair (l; r) 2 TΣ(X ) × TΣ(X ) of terms, denoted by l ! r, which satisfies the following variable conditions l 62 X , that is, the left-hand side of any rewrite rule is not a variable X + 0 ! X : OK X ! X + 0 : NG V (r) ⊆ V (l), that is, all variables appearing in r should appear in l X + s(Y ) ! s(X + Y ): OK 0 ! X ∗ 0 : NG A pair (Σ; R) of a set Σ of operators and a set R of Σ-rewrite rules is called a term rewriting system (TRS). When we call R a TRS, its Σ is the set of all operators appearing in R. The CafeOBJ reduction command ignores equations which do not satisfy the first variable conditions, e.g. X ! X + 0. NAKAMURA Masaki (Toyama Pref. Univ.) Inference with Rewriting Rules (1) December 17, 2013 6 / 22 . Subterms The set O(t) of positions of a term t is defined as O(x) = f"g and f g [ f 2 N ∗ j 2 f g 2 g O(f (¯tn)) = " i:p + i n¯ ; p O(ti ) Ex: Let t = x + s(y). O(t) = f"; 1; 2; 2:1g where " is the empty string, and ¯tn is abbr. of t1;::: tn. The subterm of a term t at position p 2 O(t), denoted by tjp, is defined as tj" = t and f (¯tn)ji:p = ti jp Ex: tj" = x + s(y); tj1 = x; tj2 = s(y); tj2:1 = y 0 The subterm relation ≥sub is defined as follows: t ≥sub t if and only 0 if t = tjp for some p 2 O(t), and the strict subterm relation >sub is defined as ≥sub n =. Ex: x + s(y) >sub s(y) >sub y NAKAMURA Masaki (Toyama Pref. Univ.) Inference with Rewriting Rules (1) December 17, 2013 7 / 22 . Substitution 0 0 The replacement of t with t at position p 2 O(t), denoted by t[t ]p, 0 0 0 0 is defined as t[t ]" = t and f (¯tn)[t ]i:p = f (:::; ti [t ]p;:::) Ex. Let t = x + s(y). t[0 + 0]1 = (0 + 0) + s(y) and t[0 + 0]2:1 = x + s(0 + 0) A map θ 2 T X from the set of variables X to a set of terms T is called a substitution if Dom(θ) = fx 2 X j θ(x) =6 xg is finite. We write θ = fx0 t0; x1 t1;:::; xn tng when θ(xi ) = ti for all i 2 Dom(θ) The instance of a term t by θ, denoted by tθ, is defined as xθ = θ(x) and f (¯tn)θ = f (tnθ). tθ0 = s(0) + s(0) if θ0 = fx s(0); y 0g tθ1 = s(y) + s(x + z) if θ1 = fx s(y); y x + zg NAKAMURA Masaki (Toyama Pref. Univ.) Inference with Rewriting Rules (1) December 17, 2013 8 / 22 . Rewrite relation For a TRS R, the rewrite relation !R is defined as follows: 0 X 0 t !l!r t () 9θ 2 T :9p 2 O(t): tjp = lθ ^ t = t[rθ]p 0 0 t !R t () 9l ! r 2 R:t !l!r t Ex. Let R+ = f(1) : X + 0 ! X ; (2) : X + s(Y ) ! s(X + Y )g s(s(0)) + s(0) !(2) s(s(s(0)) + 0) !(1) s(s(s(0))) !+ ! R is the transitive closure of R (more than zero steps) !∗ ! R is the reflexive and transitive closure of R (zero or more than zero steps). !∗ s(s(0)) + s(0) R+ s(s(s(0))) An instance of l of some l ! r 2 R is called a redex A term t is called an (R-)normal form if there is no u such that t !R u. NAKAMURA Masaki (Toyama Pref. Univ.) Inference with Rewriting Rules (1) December 17, 2013 9 / 22 . Equational reasoning ∗ A congruence relation =R is defined as (!R [ R ) 0 When we regard R as a set of equations, t =R t means the equation can be deduced from the axioms R. One of the purposes of the TRS 0 is to prove t =R t by rewriting. # 0 !∗ 0 !∗ t R t if and only if there exists u such that t R u and t R u. We call t and t0 joinalble. 0 0 It is trivial that t =R t ( t #R t 0 0 The converse t =R t ) t #R t is not always true as I showed Termination and confluence properties give us a sufficient condition 0 0 under which t =R t , t #R t NAKAMURA Masaki (Toyama Pref. Univ.) Inference with Rewriting Rules (1) December 17, 2013 10 / 22 . Termination and confluece R is terminating if there is no infinite rewrite sequence t0 !R t1 !R ··· . # 0 9 !∗ ^ !∗ 0 R is confluent if t R t whenever u. u R t u R t Termination guarantees the existence of a normal form of an input term and it can be computable in finite time, and confluence guarantees the uniqueness of normal forms. If a term t is reduced into normal forms t1 and t2, then there exists u !∗ ∗ s.t. t1 R u R t2. Since they are normal forms, we have t1 = t2. [Proposition] Let R be a terminating and confluent TRS. Then, 0 0 t =R t , t #R t NAKAMURA Masaki (Toyama Pref. Univ.) Inference with Rewriting Rules (1) December 17, 2013 11 / 22 . Proving termination Termination is undecidable, that is, there is no algorithm to solve the problem: whether R is terminating or not. There are several useful methods and tools to prove termination A basic idea is to find a well-founded ordering > on terms satisfying 0 0 t !R t implies t > t . > is well-founded if there is no infinite decreasing sequence a0 > a1 > ··· , e.g. the strict ordering on natural numbers 0 0 It is not easy to prove t > t for all t !R t since !R may be infinite (even if R is finite) For R0 = ff (X ; Y ) ! X g, ! ! we have f (a; b) R0 a and f (f (a; b); c) R0 f (a; c), and so on NAKAMURA Masaki (Toyama Pref. Univ.) Inference with Rewriting Rules (1) December 17, 2013 12 / 22 . Reduction ordering An irreflexive (t >6 t) and transitive relation is called a strict ordering A relation >⊆ T × T is stable if for all substitution θ and terms t; t0, t > t0 implies tθ > t0θ Let Σ the set of all operators. A relation >⊆ T × T is monotonic if for all f 2 Σ, ¯sn; ti 2 T , si > ti implies f (:::; si−1; si ; si+1;:::) > f (:::; si−1; ti ; si+1;:::) A rewrite ordering is a stable and monotic strict ordering A reduction ordering is a well-founded rewrite ordering [Proposition] R is terminating if and only if there exists a reduction ordering > s.t. l > r for all l ! r 2 R Two kinds of methods to obtain a reduction ordering have been considered: semantic methods and syntactic methods NAKAMURA Masaki (Toyama Pref.